No matter which of the major three airports you've arrived to, there are three main ways for you in each case to get from the airport to the city of Moscow: by car (taxi), express train and public transport.
Let's see which one you should prefer in what case. Then follow one of the links at the bottom of this article to the pages describing each mode in detail.
First of all, getting anywhere by public transport for you as a foreign tourist is supposed to be the last option only if you're in a really desperate situation or happen to be a very, very adventurous person. It's mentioned here just in case, and also because it's an actual option that exists.
The real choice for you to make is between a taxi and an express train.
1) Train brings you to the city in shortest possible time guaranteed without any delays or traffic jams. But it doesn't bring you right to the hotel doors. So you'll have you additionally use the metro or a taxi to get exactly where you need from the train terminal in the city center. Which, if you have a lot of luggage, can be quite painful.
2) A taxi takes care of all your luggage and brings you right to the point you need, but is veeery prone to stuck in a traffic jam along the way lengthening the time you're risking to spend on your trip to the city up to 2-2,5 hours in worst cases. As well as there's almost a 95% chance of significantly overpaying if you don't order it through an app.
There are two scenarios where no choice should even be on the table for you. One, when you have to get to the city ASAP without taking even a slightest chance of delay - then you take a train, and that's it. Two, is when you happen to have a strong, life-long conviction that trains are for peasants, and it doesn't matter to you how long the trip will take. Then you absolutely certainly should take a good car.
If none of the above is the case, and you're just looking for the quickest, least cumbersome and cheapest way, then you have to consider a couple factors to make the decision: traffic situation, number of people you have with you, the amount of luggage you all are carrying and how far is your hotel from the stain station on the other end.
General rules for road traffic in Moscow is that it is much better on weekends than on weekdays. And that you should avoid driving anywhere on weekdays between 6-9 AM and 6-9 PM. However, it can be absolutely unpredictable. Roads get routinely closed to government officials and visiting foreign representatives. Special events such as special street fares, marathons and concerts have a chance of shutting down entire chunks of the city. And road works and road accidents are quite ubiquitous.